Common boundary regular fixed points for holomorphic semigroups in strongly convex domains

Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb C^N$. Let $(\phi_t)$ be a continuous semigroup of holomorphic self-maps of $D$. We prove that if $p\in \partial D$ is an isolated boundary regular fixed point for $\phi_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point for $\phi_t$ for all $t\geq 0$. Along the way we also study backward iteration sequences for elliptic holomorphic self-maps of $D$.